Potential Energy Curves

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1 Potential Energy CurvesSo Far we have dealt with two kinds of potential Energy:Gravitational (U=mgh)Elastic (U=1/2kx2)Sometimes it is more helpful when you have potential energy as a function of Position to find Potential energy first and derive the force from there.

2 Force and Potential EnergyWe established last class that the relationship between work and potential energy was:which leads to…

4 What is the force when the particle is at x=-0.8m?ExampleA force parallel to the x-axis acts on a particle moving along the x-axis. The force produces a potential energy: u(x) = 1.2x4.What is the force when the particle is at x=-0.8m?

5 Force and Potential EnergyThis analysis can be extended to apply to three dimensions:

6 Check PointA particle moving along the x-axis is acted on by a conservative force. At a certain point, the force is zero.What does this tell you about the value of the potential energy function at this point?

7 Potential Energy Curves for a SpringNote:When the spring is either in a state of maximum extension or compression its potential energy is also a maximumWhen the spring's displacement is DOWN the restoring force is UPWhen the potential energy function has a negative slope, the restoring force is positive and vice-versaWhen the restoring force is zero, the potential energy is zeroAt any point in the cycle, the total energy is constant:U + K = Umax = Kmax

9 Points of EquilibriumWhen the force acting on the object is zero, the object is said to be in a state of EQUILIBRIUM!STABLE EQUILIBRIUM – located at minimums, if the object is displaced slightly it will tend back to this location.UNSTABLE EQUILIBRIUM – located at maximums, if the object is displaced slightly it will tend away from this location.STATIC EQUILIBRIUM – located at plateaus, where the net force equals zero.

10 Points of Equilibrium Stable Equilibrium at x3 and x5.Unstable Equilibrium at x4.Static Equilibrium at x1 and x6.

11 . Turning Points Define the boundaries of the particle’s motion.We know that E=K+U, so where U=E, K=0 J and the particle changes direction.For instance, if E=4J, there would be turning point at x2..

12 If E = 1J, why is the grey area referred to as an “energy well”?Turning PointsIf E = 1J, why is the grey area referred to as an “energy well”?

13 Example A particle of mass 0.5 kg obeys the potential energy function:U(x) = 2(x - 1) - (x - 2)3What is the value of U(0)? What are the values of x1 and x2? Example

14 Example A particle of mass 0.5 kg obeys the potential energy function:U(x) = 2(x - 1) - (x - 2)3How much potential energy does the particle have at position x1? If the object was initially released from rest, how fast is it moving as it passes through position x1? Example

15 Example A particle of mass 0.5 kg obeys the potential energy function:U(x) = 2(x - 1) - (x - 2)3 How much potential energy does the mass have at x2? How fast is it moving through position x2?Example

17 Example A particle of mass 0.5 kg obeys the potential energy function:U(x) = 2(x - 1) - (x - 2)3 How fast is the particle moving when its potential energy, U(x) = 0J? If x3 = ½x1, then how fast is the particle moving as it passes through position x3? Example

19 Example A particle of mass 0.5 kg obeys the potential energy function:U(x) = 2(x - 1) - (x - 2)3At what value of x does the particle experience it greatest negative acceleration? What is the value of its potential energy at this position? How much kinetic energy does it have at this position? What force is being exerted upon it at this position? What is the value of its acceleration at this position? Example